An orthogonal relation on inverse cyclotomic polynomials

Let ${\mathrm{\Phi }}_n(X)$ and ${\mathrm{\Psi }}_n(X)=\frac{X^n-1}{{\mathrm{\Phi }}_n(X)}$ be the $n$-th cyclotomic and inverse cyclotomic polynomials respectively. In this short note, for any pair of divisors $d_1\mathrm{\ne }{d}_2$ of $n$, and integers $l_1$ and $l_2$ such that $0\le l_1\le \phi (d_1)-1$ and $0\le l_2\le \phi (d_2)-1$, we show that $$⟨X^{l_1}{\mathrm{\Psi }}_{d_1}(X)(1+X^{d_1}+\dots X^{n-d_1}),X^{l_2}{\mathrm{\Psi }}_{d_2}(X)(1+X^{d_2}+\dots X^{n-d_2})⟩=0,$$ where $⟨\cdot ,\cdot ⟩$ is the inner product on $\mathbb{Q}[X]$ defined by $⟨{\sum }_k{a}_k{X}^k,{\sum }_k{b}_k{X}^k⟩={\sum }_k{a}_k{b}_k$.